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Chapter 4: Problem 2

In the United States, flow rates through showerheads are regulated to be no greater than \(9.5 \mathrm{~L} / \mathrm{min}\) under any water pressure condition likely to be encountered in a home. Water pressures in homes are typically less than \(550 \mathrm{kPa}\). A practical showerhead delivers water at a velocity of at least \(5 \mathrm{~m} / \mathrm{s}\). If nozzles in a showerhead can be manufactured with diameters of \(0.75 \mathrm{~mm}\), what is the maximum number of nozzles required to make a practical showerhead?

Short Answer

Expert verified
A practical showerhead can have a maximum of 71 nozzles.

Step by step solution

01

Calculate the Area of One Nozzle

First, find the cross-sectional area of a single nozzle using the formula for the area of a circle, \( A = \pi r^2 \). Given the diameter, \(d = 0.75\,\mathrm{mm} = 0.00075\,\mathrm{m}\), the radius \(r = \frac{d}{2} = 0.000375\,\mathrm{m}\). Thus, the area is: \[ A = \pi (0.000375)^2 \approx 4.42 \times 10^{-7} \mathrm{~m^2} \]
02

Calculate the Flow Rate per Nozzle

Using the required velocity \(v = 5\,\mathrm{m/s}\), compute the flow rate through one nozzle with \( Q = Av \). Using the area from Step 1, \[ Q = (4.42 \times 10^{-7}) \times 5 = 2.21 \times 10^{-6} \mathrm{~m^3/s} \] Convert this flow rate to liters per minute: \[ Q = 2.21 \times 10^{-6} \times 1000 \times 60 = 0.1326\,\mathrm{L/min} \]
03

Calculate the Maximum Number of Nozzles

The maximum total flow rate allowed is \(9.5\,\mathrm{L/min}\). To find the maximum number of nozzles, divide the total flow rate by the flow rate per nozzle: \[ \text{Number of Nozzles} = \frac{9.5}{0.1326} \approx 71.66 \] Since you cannot have a fraction of a nozzle, round down to 71 nozzles to ensure the flow rate does not exceed regulations.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Nozzle Design
In the context of showerheads, nozzle design plays a crucial role in determining the water's flow rate and maintaining effective water pressure. Nozzles are the small openings through which water exits the showerhead. The design parameters of these nozzles, such as diameter and shape, significantly influence how the water is distributed. The objective in designing nozzles is to provide a consistent and satisfying flow while adhering to regulations.

Key elements of nozzle design include:
  • **Diameter**: Smaller diameters increase speed and pressure.
  • **Shape**: Cone or pointed nozzles can help streamline flow, reducing splashing.
This design must strike a balance, ensuring comfort and regulatory compliance without increasing household water consumption excessively.
Velocity of Fluid
In fluid dynamics, velocity refers to how fast the fluid, like water, moves in a specific direction. For showerheads, maintaining a high enough velocity is essential to deliver a satisfying shower experience. The speed at which water exits the nozzle is determined by its velocity. A higher velocity means that water travels faster, which can provide a more powerful spray.

The velocity of water through a nozzle is governed by:
  • **Pressure**: Higher pressure generally increases velocity.
  • **Cross-sectional area**: Smaller nozzles will increase velocity for a given pressure.
For effective shower operation, designers aim to achieve a velocity of at least 5 meters per second. This velocity ensures that the water provides sufficient force to rinse effectively while still conforming to flow rate regulations.
Water Pressure Regulations
Water pressure regulations help conserve water and ensure residential plumbing systems operate within safe limits. In the United States, this regulation mandates that showerheads should not exceed a flow rate of 9.5 liters per minute, regardless of the water pressure. These regulations are in place to promote water conservation and control energy use, considering that heating water accounts for a significant part of household energy consumption.

Impact of water pressure regulations includes:
  • **Conservation**: Limits excessive water use.
  • **System stability**: Prevents leaks and breakage by operating within safe pressure ranges.
In designing a showerhead, adherence to these regulations is crucial. The challenge is to offer a satisfying shower experience while remaining within legal guidelines.
Cross-sectional Area
The cross-sectional area of a nozzle is crucial in determining how much fluid can pass through it over a given time. For a circular nozzle, this area is calculated using the formula for the area of a circle: \( A = \pi r^2 \). This calculation helps in determining flow rate and ensuring compliance with water regulations.

Understanding the cross-sectional area aids in:
  • **Flow calculation**: Determines the volume of water passing through.
  • **Design adjustment**: Smaller areas increase velocity, affecting flow characteristics.
In practical applications, knowing this area is vital to balance between sufficient flow rate and water pressure. It ensures nozzles are optimized for delivering effective water distribution while staying within permitted limits.

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Most popular questions from this chapter

A \(0.8-\mathrm{m}\) -diameter fan is driven by a \(1-\mathrm{kW}\) motor. The fan pulls in air from a large room, and air leaves the fan at \(10 \mathrm{~m} / \mathrm{s}\). Estimate the efficiency of the fan in converting electrical energy to the energy added to the air by the fan (i.e., the efficiency of the fan-motor system). Assume standard air.

Calculate the momentum-flux correction factor, \(\beta,\) for the following velocity distribution: $$ v(r)=V_{0}\left[1-\left(\frac{r}{R}\right)^{2}\right] $$ where \(v(r)\) is the velocity at a radial distance \(r\) from the centerline of a pipe of radius \(R\)

Water at \(20^{\circ} \mathrm{C}\) is contained in a pressurized tank that is supported by a cable as shown in Figure 4.71 . The (gauge) air pressure above the water in the tank is \(600 \mathrm{kPa}\), the depth of water in the tank is \(1 \mathrm{~m}\), the diameter of the tank is \(1 \mathrm{~m}\), the weight of the tank is \(1 \mathrm{kN},\) and there is a \(50-\mathrm{mm}\) -diameter orifice in the bottom of the tank. The discharge coefficient of the orifice is 0.8 . Estimate the tension in the cable supporting the tank.

Water at \(20^{\circ} \mathrm{C}\) flows through a pump in which the inflow pipe has a diameter of \(150 \mathrm{~mm}\) and the outflow pipe has a diameter of \(75 \mathrm{~mm}\). The centerlines of the inflow and outflow pipes are aligned so that there is no difference in elevation. When the flow rate through the pump is \(30 \mathrm{~L} / \mathrm{s}\), the inflow and outflow pressures are \(150 \mathrm{kPa}\) and \(500 \mathrm{kPa}\), respectively, and measurements indicate that the temperature of the water increases by \(0.1^{\circ} \mathrm{C}\) between the inflow and the outflow. Estimate the rate at which energy is being delivered by the pump to the water. Approximately what percentage of the energy input by the pump goes toward raising the temperature?

A rocket weighs \(6000 \mathrm{~kg}\), burns fuel at a rate of \(40 \mathrm{~kg} / \mathrm{s}\), and has an exhaust velocity of \(3000 \mathrm{~m} / \mathrm{s}\). Estimate the initial acceleration of the rocket and the velocity after 10 seconds. Neglect the drag force of the surrounding air and assume that the pressure of the exhaust gas is equal to the pressure of the surrounding atmosphere.
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